OPERATING CHARACTERISTICS OF AN
OPTICAL FILTER IN METALLIC
PHOTONIC BANDGAP MATERIALS
Sanjeev K. Srivastava and S. P. Ojha
Department of Applied Physics
Institute of Technology
Banaras Hindu University
Varanasi-221005, India
Received 29 March 2002
ABSTRACT: In this Letter a new analysis of an optical filter using
a metallic photonic bandgap material on the microscopic scale is
reported. The proposed filter is capable of working over a wide
range of the electromagnetic spectrum and the idea is based on the
famous Kronig–Penny model in the solid state. The periodic struc-
ture consisting of different conducting materials and dielectrics (air)
are considered. The structure is also able to pass the light emitted
by a He–Ne laser source by choosing the appropriate values of the
controlling parameters, and thus this may act as an efficient mono-
chromator. © 2002 Wiley Periodicals, Inc. Microwave Opt Technol
Lett 35: 68 –71, 2002; Published online in Wiley InterScience (www.
interscience.wiley.com). DOI 10.1002/mop.10518
Key words: optical filters; PBG materials; monochromator
1. INTRODUCTION
Recently a great deal of interest has been generated in photonic
bandgap materials because of their peculiar optical properties, such
as abnormal refractive index and gain enhancement at the band
edge [1–5]. These structures are composed of a thin dielectric
material, semiconductor material, or metallic slab surrounded by
air or other materials of lower refractive index in order to confine
the light wave. The waves incident on these materials will be
reflected if their frequency lies within the gap. The existence of the
spectral gap in such photonic crystals opens up a variety of
possible potential applications such as thresholdless semiconduc-
tor lasers [6], efficient optical filters [7, 8], omnidirectional reflec-
tors [9, 10], endlessly single-mode optical fibers, etc. [11]. Con-
ventional gratings have index modulations of few percent, whereas
PBG materials have large index contrasts in their indices to the
extent of 4:1 [12]. Due to these large index contrasts wide stop and
pass bands are obtained.
The optical filtering approach presented by Griffel [13] can be
used to design and analyze a wide range of photonic processing
networks for filtering, routing and switching. Chen et al. [14]
suggested the design of optical filters using photonic bandgap air
bridges and calculated important results regarding filtering prop-
erties. Ojha et al. [15] also suggested a method for the fabrication
of optical filters in the near- and far-infrared region. This model
was based on the weak guidance approximation such that ( n
1
-
n
2
)/ n
1
1 and the working principle is analogous to that of the
Kronig–Penny model in the band theory of solids.
In the present communication, the fabrication of an optical filter
in the visible and infrared region using the periodic refractive
index profile of the metallic conductors is suggested. There are
certain advantages to introducing metal to photonic bandgap ma-
terials. These include in the reduction of size and weight, ease of
fabrication, and lower costs. It is emphasized that the filter may
work over a wide range of the electromagnetic spectrum by the
variation of the controlling parameters, which are refractive indi-
ces and lattice parameters. This principle has been used for the
construction of monochromator.
2. THEORETICAL ANALYSIS
It is well known that when electrons move through a periodic
lattice, allowed and forbidden energy bands are obtained. The
same idea may be applicable to the case of optical radiation if the
electron waves are replaced by optical waves and the lattice
periodicity structure is replaced by a periodic refractive index
pattern. One expects allowed and forbidden bands of frequencies
(or wavelengths) instead of energies. By choosing a linearly peri-
odic refractive index profile in the filter material, one obtains a
given set of wavelength ranges that are allowed or forbidden to
pass through the filter material. Selecting a particular x axis
through the material, a periodic step function shall be assumed for
the index of the form [12, 16]
n x =
n
1
, 0 x a;
n
2
, -b x 0;
(1)
where n
1
( x + td) = n
1
and n
2
( x + td) = n
2
. Here t is the
transnational factor, which takes the values t = 0, 1, 2,
3, . . . and d = a + b is the period of the lattice, with a and b
being the width of the two regions having refractive indices ( n
1
)
and ( n
2
), respectively. In the case of a metallic conductor n
2
is a
complex quantity, and in polar form it can be written as
n
2
= + i = r cos + i sin . (2)
Thus the absolute of value of n
2
is given as
r =
2
+
2
1/ 2
, (3a)
and its amplitude
= tan
-1
. (3b)
The refractive index profile of the materials in the form of rect-
angular symmetry is shown in Figure 1.
In this case the one-dimensional wave equation for the spatial
part of the electromagnetic eigenmode
k
( x ) is given by
d
2
k
x
dx
2
+
n
2
x
2
x
c
2
k
x = 0, (4)
where n( x ) is given by Eq. (1). Assuming that the propagation is
along the z axis and n( x ) is constant in the ( n
1
) and ( n
2
) regions,
Eq. (4) for the wave equation may be written as Contract grant sponsor: University Grants Commission of India
Figure 1 Periodic variation of the refractive index profile in the form of
rectangular structure
68 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 35, No. 1, October 5 2002